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  UNIVERSIDADE NOVA DE LISBOA INSTITUTO SUPERIOR DE ESTATÍSTICA E GESTÃO DE INFORMAÇÃO Introduction to Kohonen’s Self-Organizing Maps   Fernando Bação and Victor Lobo    Introduction to Kohonen’s Self-Organizing Maps Fernando Bação and Victor Lobo 2 CONTENTS 1 - INTRODUCTION..................................................................................................3   2 – THE ORIGINS OF THE SELF-ORGANIZING MAP......................................3   3 - BASIC PRINCIPLES.............................................................................................4   4 - FORMAL DESCRIPTION OF THE TRAINING ALGORITHM.................13  4.1   -   T HE ALGORITHM ................................................................................................13 4.2   -    N EIGHBORHOOD FUNCTIONS ..............................................................................14 4.3    –    A   S IMPLE E XAMPLE OF HOW THE SOM  WORKS ................................................15 5 – ADDITIONAL FEATURES AND COMMENTS.............................................17   7 - REFERENCES.....................................................................................................21      Introduction to Kohonen’s Self-Organizing Maps Fernando Bação and Victor Lobo 3 1 - Introduction This text is meant as a tutorial on Kohonen’s Self-Organizing Maps (SOM). The basic idea is to provide an overview of this valuable tool, allowing the students to understand the basic principles of its workings. Clearly, this objective can only be achieved if the student background is considered, for this reason we will try to easy the technical aspects through the use of simple working examples. Emphasis will be  put on practical work, hopefully motivating the student to continue on his own the study of the SOM. We shall start with a short introduction to SOM, explaining what it is. We will attempt to give an intuitive approach to its underlying principles, and  proceed to a more formal definition of SOM. We would like to emphasize the importance of the accompanying materials in order to get a “complete” picture of the SOM and its potential within GISc. This way we encourage the students to read the papers suggested in the web page of the course. Additionally, the demos can be valuable elements in order to develop an intuitive understanding of the SOM workings. 2 – The Origins of the Self-Organizing Map Although the term “Self-Organizing Map” could be applied to a number of different approaches, we shall always use it as a synonym of Kohonen’s Self Organizing Map, or SOM for short. These maps are also referred to as “Kohonen Neural Networks” [Fu 94], “Self-organizing Feature Maps-SOFM”, or “Topology preserving feature maps” [Kohonen 95], or some variant of these names. Professor Kohonen worked on auto-associative memory during the 70’s and early 80’s, and presented his self-organizing map algorithm in 1982 [Kohonen 82]. However, it was not until the publication of the second edition of his book “Self-    Introduction to Kohonen’s Self-Organizing Maps Fernando Bação and Victor Lobo 4 Organization and Associative Memory” in 1988 [Kohonen 88a], and his paper named “The Neural Phonetic Typewriter” on IEEE Computer [Kohonen 88b] that his work  became widely known. Since then there have been many excellent papers and books on SOM, but his 1995 and 2001 books [Kohonen 95, 01] are generally regarded as the major references on the subject. The books have had very flattering reviews,  presenting a thorough covering of the mathematical background for SOM; its  physiological interpretation; the basic SOM; and recent developments and applications. Professor Kohonen’s group maintains a very good web-site at Helsinki’s Technical University at “ http://www.cis.hut.fi/research/ ”. That site contains public domain software, various manuals, papers, technical reports, and a very thorough and searchable list of papers dealing with SOM and LVQ. We strongly recommend a visit to that site to anyone that intends to work on or with SOM. Kohonen himself describes SOM as a “visualization and analysis tool for high dimensional data”. These are indeed two of the most attractive characteristics of the SOM, but it can also be used for clustering, dimensionality reduction, classification, sampling, vector quantization, and data-mining. Here we will focus on the use of the SOM as an exploratory analysis tool, emphasizing the aspects related with clustering, visualization and the analysis of high dimensional data. Our perspective is that the SOM can be viewed as a toolbox rather than a tool, as it contains numerous features that can be of interest in different situations and for different tasks. 3 - Basic principles The SOM encompasses a number of characteristics which bear similarities to the way the human brain works, or at least is thought to work. In fact, the notion of having available a set of neurons which through learning experiences specialize in the identification of certain types of patterns is consistent with current research on the human brain. The idea that certain parts of the brain are responsible for specific skills
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